Is fuzzy logic connected to neural networks?
They are unrelated.

There is a possibility of interpreting fuzzy values as probabilities, but strictly speaking they are different: fuzzy values are vague, while probabilities reflect likelihood (see Wikipedia entry for Fuzzy Logic)

While rolling a particular number on a six-side die has a probability of $1 \over 6$, a roll can actually only ever have one outcome.

A fuzzy value "quite old" can simultaneously be member of a number of fuzzy sets with different degrees of membership, eg "young" with 0.001, "adolescent" with 0.1, "old" with 0.4, "ancient" with 0.7. Unless it is "defuzzified", it is simultaneously contained in all the sets.

Defuzzyfication is a way of interpreting the result of a series of fuzzy operations and finding the set that best matches, but it is not a clearly defined process such as picking a random number according to a set of probabilities (or rolling the die).

I am not sure that the sum of all fuzzy set membership values of any given fuzzy value has to add up to 1.0; whereas this condition has to hold for probabilities.

[EDIT: to clarify - probabilities are not a set; I refer here to all possible outcomes of a random event which have a certain probability of being realised. The sum of all possible event probabilities has to be 1.0]

One alternative interpretation for your application could be the confidence that the input set is identical to the training set. Which could be a fuzzy value if you wanted to do something else with it, eg by combining it with other fuzzy variables.